I like O2 and it's easy to keep their SIMs active.
Regarding the perception that cellular data service is slow with any particular carrier, double-check whether your plan subjects you to "throttling" (speed reduction after you've used a certain amount of data), and whether your phone supports all (or as many as possible) of the LTE bands that your carrier uses in the countries you are visiting. Use of the radio frequency spectrum has been evolving rapidly over the past year or two, and changes will continue. If your phone is a few years old, replacing it during this model year or the next might be beneficial. Check detailed specifications online and match the exact part number on the phone's box. Apple, for example, has simultaneously offered multiple versions of the iPhone in the US, with slight differences in LTE band lineups.
Re: buying in London, enjoy free roaming while it lasts! Now that Boris has a solid majority, Brexit is back on. Post-Brexit, the EU rule that forbids carriers from charging for roaming won't apply to UK SIMs used in EU countries, or to EU SIMs used in the UK.